ABSTRACT
For initial conditions that are captured into the right well, there is a sequence of double pulses (each composed of a right solitary pulse followed by a left solitary pulse) followed by a sequence of captured right solitary pulses, as sketched in Figure 2. We designate t0 as the unknown time of the last saddle approach with positive energy, W0 > 0. Since the energy will diminish by fDR on the next right solitary pulse, the parameter Wo must satisfy
(29)
(30)
The parameter 7/Jc, and hence 7/J(jod, is known from initial conditions (in a sensitively dependent way), so that (30) actually determines W0 from initial conditions (in a sensitively dependent way). Since for right capture 0 < W0 < EDR, it follows from (30) that
(32)
1 fTc 1 {E(o) dE '1/Jc = '1/J(Tc) = ~ lo wo dT + cf>(Tc) = ~ lo Do( E)+ cf>(Tc), (33)
where N = ['1/Yc] is a large 0(1/E) integer and the boundaries of the basin of attraction for the right well correspond to Wo = 0 and W0 = EDR. Only the slow variation equations are used (with our observations and Melnikov energy considerations) to determine the phase for the slow variation equations (32) corresponding to the boundaries of the basin of attraction (the perturbed stable manifold of the saddle point). We note from (34) (let ~ W 0 = EDR) the previously known elementary result, that the difference in initial energies corresponding to right capture is the appropriate fraction (DR/(DR + DL)) of the initial energy dissipation: EDo(E(O))DR/(DR + DL)·
pulse. These solitary pulses are symmetric about their centers. In this section, we explain how we determined the distance between successive solitary pulses [3].
